Question

The image shows a composite shape made up of a rectangle and a triangle. The rectangle has a width of 4 cm and a height of 6 cm. The triangle has a base of 4 cm (same as the rectangle's width) and a side of 5 cm. The other side of the triangle is 9 cm.

Answer 1

To calculate the area of the composite shape made up of a rectangle and a triangle, we will first find the areas of each individual shape and then add them together.

### Step 1: Calculate the area of the rectangle.
The formula for the area of a rectangle is:
\[
\text{Area} = \text{width} \times \text{height}
\]

For the rectangle given:
- Width = 4 cm
- Height = 6 cm

Now, calculating the area:
\[
\text{Area}_{\text{rectangle}} = 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2
\]

### Step 2: Calculate the area of the triangle.
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here we need to determine the height of the triangle. We know:
- Base \(b = 4 \, \text{cm}\)
- One side \(a = 5 \, \text{cm}\)
- The other side \(c = 9 \, \text{cm}\)

Since we have all three sides, we can use Heron's formula to find the area of the triangle. First, we need to calculate the semi-perimeter \(s\):
\[
s = \frac{a + b + c}{2} = \frac{5 \, \text{cm} + 4 \, \text{cm} + 9 \, \text{cm}}{2} = 9 \, \text{cm}
\]

Now apply Heron's formula:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]

Calculating each term:
- \(s - a = 9 \, \text{cm} - 5 \, \text{cm} = 4 \, \text{cm}\)
- \(s - b = 9 \, \text{cm} - 4 \, \text{cm} = 5 \, \text{cm}\)
- \(s - c = 9 \, \text{cm} - 9 \, \text{cm} = 0 \, \text{cm}\)

So, we have:
\[
\text{Area} = \sqrt{9 \, \text{cm} \times 4 \, \text{cm} \times 5 \, \text{cm} \times 0 \, \text{cm}} = \sqrt{0} = 0 \, \text{cm}^2
\]

This indicates that the triangle does not actually form a valid shape (it is degenerate) because one side equals the sum of the other two sides. Therefore, the height is 0 and the area is effectively 0.

### Step 3: Combine the areas of the two shapes.
The total area of the composite shape is:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{triangle}} = 24 \, \text{cm}^2 + 0 \, \text{cm}^2 = 24 \, \text{cm}^2
\]

### Final result:
The area of the composite shape is \(24 \, \text{cm}^2\).